in Graph Theory retagged by
8,015 views
0 votes
0 votes

If $G$ is an undirected planar graph on $n$ vertices with $e$ edges then

  1. $e\leq n$
  2. $e\leq 2n$
  3. $e\leq 3n$
  4. None of the option
in Graph Theory retagged by
by
8.0k views

4 Answers

3 votes
3 votes

for planar graphs, 3r <= 2e and v - e + r = 2

From these we get  e <= 3v - 6

Now 2v <= 3v - 6 for all vertices v >=3

So e <= 2n is the answer.

1 vote
1 vote

Theorem 1 (Euler's Formula)   Let $G$ be a connected planar graph, and let $n, m$ and $f$ denote, respectively, the numbers of vertices, edges, and faces in a plane drawing of $G.$ Then $n - m + f = 2.$

Corollary 1    Let $G$ be a connected planar simple graph with $n$ vertices, where $n \geq 3$ and $m$ edges. Then $m \leq 3n - 6.$

Corollary 2     Let $G$ be a connected planar simple graph with $n$ vertices and $m$ edges, and no triangles. Then $m \leq 2n - 4.$

Reference 

edited by
0 votes
0 votes
0 votes
0 votes
A planar graph of order n and size m. Then, m ≤ 3n-6.
Answer:

Related questions

Quick search syntax
tags tag:apple
author user:martin
title title:apple
content content:apple
exclude -tag:apple
force match +apple
views views:100
score score:10
answers answers:2
is accepted isaccepted:true
is closed isclosed:true